A well completion engineer needs a variety of information in order to optimally complete a well. Two of the most important pieces of information relative to well completion include the ratio of near field permeability to far field permeability, and the radial depth of the wellbore damage zone. If known, these two factors may enable a completion engineer to enhance production. Permeability is sometimes expressed in terms of fluid viscosity (i.e. a ratio of permeability to viscosity). The ratio of permeability to viscosity is generally referred to as “mobility.” The radial depth of the wellbore damage zone may be determined by generating horizontal and vertical shear slowness profiles. Many logging tools and procedures have been developed over the years to aid with the formation evaluation process, including estimates of formation mobility.
One example of a logging device that has been used to obtain and analyze sonic logging measurements of formations surrounding an earth borehole is Schlumberger's Modular Sonic Imaging Platform (MSIP) tool. According to conventional use of the MSIP logging tool, one can present compressional slowness, Δtc, shear slowness, Δts, and Stoneley slowness, Δtst, each as a function of depth, z (slowness corresponds to the interval transit time typically measured by sonic logging tools).
An acoustic source in a fluid-filled borehole generates headwaves, as well as relatively stronger borehole-guided modes. A standard sonic measurement system includes a piezoelectric source and hydrophone receivers inside a fluid-filled borehole. The piezoelectric source may be either a monopole or a dipole source. The source bandwidth typically ranges from a 0.5 to 20 kHz. A monopole source primarily generates the lowest-order axisymmetric mode, also referred to as the Stoneley mode, along with compressional and shear headwaves. In contrast, a dipole source primarily excites the lowest-order flexural borehole mode together with compressional and shear headwaves. The headwaves are caused by the coupling of the transmitted acoustic energy to plane waves in the formation that propagate along the borehole axis. An incident compressional wave in the borehole fluid produces critically refracted compressional waves in the formation. The waves refracted along the borehole surface are known as compressional headwaves. The critical incidence angle is represented as θi=sin−1(Vf/Vc), where Vf is the compressional wave speed through the borehole fluid and Vc is the compressional wave speed through the formation. As a compressional headwave travels along an interface, it radiates energy back into the fluid that can be detected by the hydrophone receivers placed in the fluid-filled borehole. In relatively fast formations, the shear headwave can be similarly excited by a compressional wave at the critical incidence angle θi=sin−1(Vf/Vs), where Vs is the shear wave speed through the formation. It is also worth noting that headwaves are excited only when the wavelength of the incident wave is smaller than the borehole diameter so that the boundary can be effectively treated as a planar interface. In a homogeneous and isotropic model of fast formations, as above noted, compressional and shear headwaves can be generated by a monopole source placed in a fluid-filled borehole to determine the formation compressional and shear wave speeds. However, refracted shear headwaves cannot be detected for slow formations (where the shear wave velocity is less than the borehole-fluid compressional wave velocity) with receivers placed in the borehole fluid. Therefore, formation shear velocities are obtained from the low-frequency asymptote of flexural dispersion for slow formations. There are standard processing techniques for the estimation of formation shear velocities in either fast or slow formations from an array of recorded dipole waveforms.
In a recently issued patent (U.S. Pat. No. 6,611,761 for “Sonic Well Logging for Radial Profiling,” hereby incorporated by this reference) a technique is described for obtaining radial profiles of fast and slow shear slownesses using measured dipole dispersions in the two orthogonal directions that are characterized by the shear moduli c44 and c55 for a borehole parallel to the X3-axis 102 (FIG. 1, the X3-axis is a borehole 100 axis). The meanings of shear moduli c44 and c55 are generally known to those of skill in the art and are also defined below.
Another U.S. patent (U.S. Pat. No. 6,714,480 entitled “Determination of Anisotropic Moduli of Earth Formations” hereby incorporated by this reference) describes a technique for estimating the horizontal shear modulus c66 (c66 is generally known to those of skill in the art and is also defined below) of a TIV (transverse isotropy with a vertical axis of symmetry) formation using the zero frequency intercept of the Stoneley dispersion that yields the tube wave velocity. This technique assumes that the borehole Stoneley dispersion is insignificantly affected by the presence of the sonic tool structure or any possible near-wellbore alteration, such as super-charging in permeable formation, and shale swelling in overburden shales. However, recent observations reveal that in fast formations and boreholes with small diameters, both the sonic tool effect and near-wellbore alteration can have significant effects on the measured Stoneley dispersion.
To overcome these limitations in the estimation of horizontal shear modulus c66 in the far-field of a TIV-formation with the TI-symmetry X3-axis parallel to the borehole, principles described below provide procedures for obtaining radial profiles of horizontal shear slowness and estimates of the horizontal shear modulus c66 outside any near-wellbore altered annulus. The far-field shear modulus c66 may then be appropriately used to characterize formation TIV-anisotropy for subsequent application in the AVO (amplitude verses offset) analysis.
According to the prior art, formation mobility is estimated from an increase in Stoneley slowness (and or attenuation) over a certain bandwidth as the Stoneley wave propagates through a permeable interval. FIG. 1A is a schematic diagram of the deformation of the fluid-filled borehole 100 caused by the lowest-order axi-symmetric wave propagating along the borehole 100. The Stoneley mobility indicator is based on the difference between measured Stoneley slowness and theoretical Stoneley slowness for a non-permeable interval (S—Se). Chang et al., Low-frequency tube waves in permeable rocks, GEOPHYSICS, Vol. 53 No. 4, pp. 519-527 (April 1988).
However, there are a number of drawbacks associated with using this Stoneley mobility indicator to estimate permeability (or, equivalently, mobility). The primary disadvantage of using the Stoneley mobility indicator is misinterpretation of permeability zones. Changes in formation shear modulus caused by formation stresses or varying amounts of clay content can be misinterpreted as zones of higher permeability. This technique assumes that the hear modulus c66 derived from the Stoneley data and the shear modulus (c44 or c55) derived from the dipole data are identical in the absence of formation mobility (as would be the case in an isotropic formation). FIG. 1B illustrates theoretical Stoneley and flexural dispersions for a range of formation slownesses. The illustrated results represent an isotropic and homogeneous formation in the absence of any tool effects. Consequently, radial profiles of horizontal shear slowness from the Stoneley dispersion data and vertical shear slowness from the dipole dispersion data would essentially overlay in the absence of any formation mobility. Any observed separation between these profiles in the far-field is taken as the Stoneley mobility indicator.
The general expression yielding the Stoneley slowness in this low-frequency limit is (Chang et al. (1988); Liu et al., Effects of an Elastic Membrane on Tube Waves in Permeable Formations, J. ACOUST. SOC. AM., Vol. 101 No. 6, pp. 3322-3329 (June 1997)):
                              S          2                =                              ρ            m                    ⁡                      (                                                                                1                                          K                      m                                                        +                                      1                    G                                                  ︸                            +                                                2                                      a                    ⁡                                          (                                                                        W                          o                                                +                                                  W                          mc                                                                    )                                                                      ︸                                      )                                              (        1        )            
Elastic Frequency and Biot Effects
where
                                          W            0                    =                                    -                              μ                k                                      ⁢                          C              D                        ⁢                          k                              c                2                                      ⁢                                                            H                  0                                      (                    l                    )                                                  ⁡                                  (                                                            k                                              c                        ⁢                                                                                                  ⁢                        2                                                              ⁢                    a                                    )                                                                              H                  1                                      (                    1                    )                                                  ⁡                                  (                                                            k                                              c                        ⁢                                                                                                  ⁢                        2                                                              ⁢                    a                                    )                                                                    ⁢                                  ⁢        and                            (        2        )                                          k                      c            2                          =                              i            ⁢                                                  ⁢                          ω              /                              C                D                                                                        (        3        )            with                S Measured Stoneley slowness        k Matrix Permeability        η Pore fluid viscosity        ρm Mud or borehole fluid density        Km Mud or borehole fluid bulk modulus        G Isotropic Formation shear modulus where c44=c55=c66        a Borehole radius        ω Angular frequency (2πf)        kc2 The Biot slow-wave wavenumber        CD Slow wave diffusivity        Wo Acoustic fluid mobility effects        Wmc Acoustic mud cake membrane stiffness        Hj(1)(x) (outgoing) Hankel functions, j=0,1The slowness, S, is complex-valued and frequency dependent; the real part of S is the conventional phase slowness and the imaginary part is simply related to the attenuation. The term        
      2    ⁢    ρ        a    ⁡          (                        W          o                +                  W          mc                    )      represents the contribution of fluid mobility effects to the Stoneley slowness. The quantity W0, which is a function of frequency, describes the permeability dependence. The membrane stiffness parameter, Wmc, describes how hard it is to force fluid in through the borehole wall, past the mudcake, which is assumed to flex in and out under the effect of an oscillating pressure. Large values of Wmc imply that the mudcake seals the borehole. Intuitively, in such cases, the slowness and attenuation are not sensitive to permeability, and the low-frequency formula bears this out. No mudcake corresponds to Wmc=0, which maximizes the sensitivity of the Stoneley to permeability effects. However, as illustrated in FIG. 2, a permeable interval 104 affects Stoneley waves. The permeable interval 104 increases Stoneley slownesses at all frequencies and introduces attenuation of the Stoneley wave amplitude.
The S—Se technique derives from the approximation proposed by White. White, J. E., Underground Sound—Application of Seismic Waves, Chapter 5—Waves Along Cylindrical Boreholes, ELSEVIER, New York (1983), pp. 139-191. The general low-frequency equation can be written as:
                              S          2                =                              S            e            2                    +                                    2              ⁢                              ρ                m                                                    a              ⁡                              (                                                      W                    o                                    +                                      W                    mc                                                  )                                                                        (        4        )            FIG. 3 shows a correlation between (S—Se) and formation mobility for a few different values of mud compressional slowness. This type of correlation shows that one can infer formation mobility from the estimated (S—Se) within the framework of low-frequency approximation as discussed by Chang et al., 1988.
It would be useful to have accurate and quantitative radial profiles of the three shear slownesses characterized by the shear moduli c44, c55, and c66 in the three orthogonal coordinate planes that can be employed in the evaluation of formations for the presence near-wellbore permeability impairment and/or producibility of hydrocarbons. Therefore, principles of the present invention provide methods and apparatus for radial profiling of formation mobility, and radial profiles of formation mobility aid in the optimal design of a well completion.